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D 102, 064021 (2020)

Critical behavior of charged holes in AdS space

Amin Dehyadegari1 and Ahmad Sheykhi1,2,3,* 1Physics Department and Biruni Observatory, Shiraz University, Shiraz 71454, Iran 2Research Institute for and of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran 3Max-Planck-Institute for Gravitational (Albert-Einstein-Institute), 14476 Potsdam, Germany

(Received 26 February 2020; accepted 21 August 2020; published 9 September 2020)

We revisit critical behavior and structure of charged anti–deSitter (AdS) dilaton black holes for arbitrary values of dilaton coupling α, and realize several novel phase behavior of this system. We adopt the viewpoint that cosmological constant () is fixed and treat the charge of the black hole as a thermodynamical variable. We study critical behavior and phase structure by analyzing the phase diagrams in T − S and q–T planes. We numerically derive the critical point in terms of α and observe that for α ¼ 1 pffiffiffi and α ≥ 3, the system does not admit any critical point, while for 0 < α < 1, the critical quantities are not α significantly affected by . We find that unstable behavior of the Helmholtz free energy for qqc, however, a novel first order phase transition occurs between small and large black hole, which has not been observed in the previous studies on phase transition of charged AdS black holes.

DOI: 10.1103/PhysRevD.102.064021

I. INTRODUCTION Recently, it has been shown that for charged AdS black hole a second and first order phase transitions occur Phase transition is certainly one of the most intriguing between small and large black holes in an extended phase and interesting phenomena in the thermodynamic descrip- space which resembles the liquid-gas phase transition in the tion of black holes which may shed on the of usual van der Waals liquid-gas system [3,4]. In an extended quantum . In particular, the investigations on the thermodynamic phase space, the cosmological constant black hole phase structure/transition in an asymptotically (AdS length) is considered as a thermodynamic pressure AdS have received considerable attentions in the which can vary, and its corresponding conjugate quantity is past years. This is mainly due to the remarkable duality the thermodynamic volume of the black hole. Taking into between gravity in an (n þ 1)-dimensional AdS spacetime account the variation of pressure in the first law, one and the conformal theory living on the boundary of observes that the of AdS black hole is equivalent its n-dimensional spacetime, (AdS=CFT) correspondence. to the enthalpy [5]. In the recent years, thermodynamic Perhaps, one of the earliest studies in this direction was phase transitions in an extended phase space have been done by Hawking and Page [1], who disclosed that there is explored for various types of black holes in AdS space (see indeed a first order phase transition, latter named Hawking- Refs. [6–22] and references therein). Reference [23] per- Page phase transition, between the and formed a stability analysis of charged AdS dilaton black the stable large Schwarzschild black hole with spherical hole by considering the second derivative of mass with horizon in the background of AdS spacetime. Later, Witten respect to . It was shown that the black holes are discovered [2] that this phase transition can be interpreted thermodynamically stable for small α, while for large α in the AdS=CFT duality as the confinement/deconfinement black hole becomes unstable [23]. The studies on phase phase transition of the strongly coupled . transition in such black hole have been carried out from both thermal and dynamical point of view [24], where the *[email protected] cosmological constant appears as a thermodynamical variable. They found that for small dilaton coupling, Published by the American Physical Society under the terms of α ≈ 0.01, the system resembles the van der Waals fluid the Creative Commons Attribution 4.0 International license. behavior, while for α > 1, the P − v diagram of the system Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, deviates and new phenomena beyond the van der Waals and DOI. Open access publication funded by the Max Planck liquid-gas-like appears [24]. Recently, a novel phase Society. behavior represents a small/large black hole zeroth-order

2470-0010=2020=102(6)=064021(9) 064021-1 Published by the American Physical Society AMIN DEHYADEGARI and AHMAD SHEYKHI PHYS. REV. D 102, 064021 (2020) phase transition, in an extended phase space with varying such that black hole has different behaviors for 0 < α < pffiffiffi pffiffiffi cosmological constant, has been observed for charged 1= 3 and 1= 3 < α. Besides, we calculate the critical dilaton black holes, where the geometry of spacetime is point numericallyffiffiffi for various values of α in T-S plane. When not asymptotically AdS [25]. p α ¼ 1 and α ≥ 3, we cannot realize any critical point in the Another possible approach to study thermodynamic system which has not been reported in the previous inves- phase structure of black hole is to consider the variation tigations on phase transition of charged AdS black holes. of of the black hole, while the cosmological Besides, for 0 ≤ α < 1, we observe a small/large first constant (AdS length) is kept fixed. With regard to this order phase transition for qq. Indeed, there is a charged AdS black holes were investigated in a fixed AdS c pffiffiffi 1 α 3 geometry, indicating that it exhibits the small/large black novel first order phase transition in the range of < < hole phase transition of van der Waals type [26–28]. which has not previously been observed. Finally, the phase Interestingly enough, it has been realized [26] that the diagram for charged AdS dilaton black hole is constructed phase transition of charged AdS black hole can occur in in q-T plane. 2 − Ψ Ψ 1 2 2 This paper is structured in the following manner. In Q plane, where ¼ = rþ is the conjugate of Q , without extending the phase space. Indeed, in this alter- Sec. II, a brief overview on of the four- native perspective, the relevant response function clearly dimensional charged dilaton black hole in the AdS back- signifies the stable and unstable region. Also there still exist ground is given. In Sec. III, we investigate critical behavior of charged dilaton AdS black holes by studying the specific a deep analogy between critical phenomena and critical heat at constant electric charge in T-S plane. In Sec. IV,we exponents of the system with those of van der Waals liquid- use the Helmholtz free energy to determine the possible gas system [26]. The advantages of this new approach is phase transition in the system. Finally, we summarize the that one do not need to extend the phase space by treating main results Sec. V. the cosmological constant as a thermodynamical variable which may physically make no sense [26]. It has been confirmed that this new approach also works in other II. CHARGED DILATON BLACK HOLES gravity theories [29,30] as well as in higher spacetime IN AdS SPACE dimensions [31]. Recently, the universality class and We start with a brief review on charged AdS black holes critical properties of any AdS black hole, independent of in dilaton gravity and calculate the associated conserved spacetime metric, via an alternative phase space has been and thermodynamic quantities. Exact charged dilaton black explored [32]. It was shown that the values of critical hole solutions in the background of (A)dS space in four exponents for generic black hole are the same as ones of [35] and higher dimensions [36,37] have been presented the van der Waals fluid system [32]. For Born-Infeld AdS by Gao and Zhang. Thermodynamics of these solutions black holed in four dimensional spacetime, we analytically has been investigated in [23,38]. The four-dimensional calculated the critical point by studying the behavior of action of Einstein-Maxwell gravity coupled to a dilaton specific heat in a fixed AdS geometry [33]. Furthermore, field is [23,35] the interesting reentrant phase transition of Born-Infeld Z AdS black hole has been investigated in the thermodynamic 1 ffiffiffiffiffiffi S − 4 p− R − 2 ∇φ 2 − φ phase space [33]. The structure of charged black holes has ¼ 16π d x gð ð Þ Vð Þ been explored by employing the Ruppeiner geometry [34] −2αφ μν − e FμνF ; 1 in which the charge is allowed to vary. Þ ð Þ In this paper, we study thermodynamic properties of where R is the Ricci scalar curvature, φ is the dilaton field charged AdS dilaton black hole including the phase tran- and VðφÞ is the dilaton potential. Herein, the electromag- sition and critical behavior in (3 þ 1)-dimensional space- netic field tensor Fμν is defined in terms of the gauge field . Our work differs from [24] in that we keep the ∂ − ∂ cosmological constant as a fixed quantity and treat the Aμ via Fμν ¼ μAν νAμ. For an arbitrary value of the α charge of the black hole as a thermodynamic variable, while dilaton coupling strength in AdS space, the dilaton the authors of [24] extended the phase space by treating potential is chosen to take the following form [35,39] cosmological constant as a variable. Following [33],wefirst 2Λ 2 identify different types of charged AdS dilaton black hole V φ 8α2eðα −1Þφ=α − α2 − 3 e2αφ ð Þ¼3 α2 1 2 ½ ð Þ depending on the behavior of at constant ð þ Þ charge for arbitrary values of the dilaton-electromagnetic þ α2ð3α2 − 1Þe−2φ=α; ð2Þ coupling constant α and fixed AdS length (cosmological constant). This has not been addressed in the previous where Λ is the cosmological constant that relates to the AdS studies [23,24]. As we shall see, the behavior of black hole radius l as Λ ¼ −3=l2. The potential given in Eq. (2) shows crucially depends on α for the small entropy, that the cosmological constant Λ is coupled to the dilaton

064021-2 CRITICAL BEHAVIOR OF CHARGED DILATON BLACK HOLES … PHYS. REV. D 102, 064021 (2020) field φ in a nontrivial way. When the coupling constant as the Hawking temperature T, entropy S, and electric pffiffiffi pffiffiffi α ¼1= 3; 1; 3, the dilaton potential in Eq. (2) is potential U, are indeed the (SUSY) potential of    1 b 1−γ ρ theory. Note that in the absence of the dilaton field, T ¼ 1 − 1 þ þ ½3ρ þ 2bðγ − 2Þ i.e., V φ 0 2Λ, the action Eq. (1) reduces to the 4πρ ρ l2 þ ð ¼ Þ¼ þ  þ  usual Einstein-Maxwell theory with cosmological constant. b 2ðγ−1Þ 3 1 1 − In þ dimensions, the line element of a static spherically × ρ ; ð10Þ symmetric spacetime is written þ   ρ2 ρ2 b γ q 2 2 d 2 2 2 þ 1 − ds ¼ −fðρÞdt þ þ ρ R ðρÞdΩ ; ð3Þ S ¼ 4 ρ ;U¼ ρ ; ð11Þ fðρÞ þ þ 2 where dΩ is the metric of the 2-dimensional unit sphere where entropy S is written per unit volume ω. It is easy to with volume ω ¼ 4π and the metric functions fðρÞ and verify that the first law of black hole thermodynamics RðρÞ are given by [23]        2 dM TdS UdQ; b γ b 1−2γ c ρ ¼ þ ð12Þ fðρÞ¼ 1 − 1 − 1 − þ ; ð4Þ ρ ρ ρ l2   is satisfied on the horizon [23]. γ It is worthwhile to mention that in the absence of the 2 ρ 1 − b R ð Þ¼ ρ ; ð5Þ dilaton filed (α ¼ 0), the solutions reduce to the well-known four-dimensional Reissner-Nordstrom (RN)-AdS black hole. where b and c are integration constants and γ ¼ It is also notable to mention that these solutions are even 2 2 2α =ðα þ 1Þ. Also, the dilaton field and the only non- functions in α. In the next section, we study the critical vanishing component of the gauge field Aμ are obtained behavior of dilaton AdS black hole in the phase space. as [23] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   γ 2 − γ III. CRITICAL BEHAVIOR OF CHARGED φ ρ ð Þ 1 − b − q ð Þ¼ 2 ln ρ ;At ¼ ρ ; ð6Þ DILATON AdS BLACK HOLE In this section we are going to investigate the effects of where q, an integration constant, is the charge parameter the dilaton field on the critical behavior of charged dilaton which is related to b and c via the following relation AdS black hole. To end this, we analyze behavior of the bc specific heat at constant charge q2 ¼ : ð7Þ α2 þ 1   α ≠ 0 dS For , these solutions become imaginary in the range of Cq ¼ T ; ð13Þ 0 < ρ

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constant charge is positive. On the other hand, the 0.30 large entropy limit of the temperature is 0.25 pffiffiffi 0.20 0 ≈ 3 S ⇒ 2 0 T 2π 2 Cq ¼ S> ; ð19Þ T 0.15 0 1 3 l 1 3 0.10 1 3 1 which is independent of the charge and dilaton 0.05 1 coupling constant and always yields a thermal stable 1 0.00 system. The thermal stability of a charged AdS dilaton 0.05 ∂2 ∂ 2 0.00 0.05 0.10 0.15 0.20 0.25 0.30 black hole was analyzed by means of ð M= S Þq S for higher dimensional at the fixed l and b in Ref. [23]. By fixing l and b, the charge of FIG. 1. T-S diagram of charged dilaton AdS black hole. This black hole is determined as a function of ρ and α α þ figure shows the remarkable influence of the coupling constant using the black hole horizon relation fðρ Þ¼0 on the temperature. Here, we have set l 1 and q 1. þ ¼ ¼ and Eq. (7), pffiffiffi ρ α (ii) For α ¼ 1= 3, q ¼ qð þ; Þ: ð20Þ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3l2 þ 12q2 − l 48q2 þ 9l2 Indeed, charge (q) is not fixed in the corresponding pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  figures of Ref. [23]. Here, we classify the four T ¼ 3=2 2πl4q2 9 þ 48q2=l2 − 3 dimensional charge AdS dilaton black holes by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi examining the behavior of the specific heat, Eq. (13), at fixed l (i.e., cosmological constant) 3l2 8q2 l 48q2 9l2S O S3 ; × þ þ þ þ ð Þ and q, for various values of α. Such classification ð15Þ was not performed in previous studies [23,24]. In what follows, we are going to obtain the critical where the dilaton black hole has zero temperature at point, which corresponds to a second order phase the vanishing entropy limit. This α may be called the transition, for various type of charged dilaton AdS “ ” α black holes. For fixed q and l, the value of the critical marginalpffiffiffi coupling constant ( m). (iii) For 1= 3 < α < 1, point is characterized by the inflection point

2 2 2 2 1=ð2α Þ−1 ∂T ∂ T ð3α − 1Þðα þ 1Þ 2 0 0 1=ð2α Þ−1=2 ¼ ; 2 ¼ : ð21Þ T ¼ 2 2 2 þ OðS Þ; 1=α þ1 2 −1=α 1=ð2α Þ−1=2 ∂S ∂S π2 l q S qc qc ð16Þ To calculate the above expressions from Eq. (10) and (11), the following relation is used black hole is “Schwarzschild-AdS” (Schw)-type. In

this case, black hole solution does not exist in the ∂T ∂T ∂b ∂ρ þ ∂ ∂ρ low-temperature regime. ∂T þ b ρ þ b;q þ;q q α 1 ¼ ; (iv) For ¼ , ∂S ∂S ∂S ∂b q ∂ρ þ ∂ ∂ρ þ b ρ þ b þ q l2 þ 2q2 T ¼ pffiffiffi þ OðSÞ; ð17Þ 4 2l2πq where

“ ” ∂ ρ which is the spatial case where the dilaton black fð þÞ ∂ ∂ρ hole has finite temperature at S 0. b þ b;q ¼ ¼ − : ∂ρ ∂fðρ Þ (v) For α > 1, þ q þ ∂b ρ þ;q 21=α2−3 α2 1 −1=ð2α2Þ ð þ Þ 1=2−1=ð2α2Þ Due to the complicated form of Eqs. (10) and (21),it T ¼ 2 2 þOðS Þ; ð18Þ πq1=ðα ÞS1=2−1=ð2α Þ is almost impossible to obtain the critical values, analytically. Hence, we numerically solve the set black hole is Schw-type again. As can be seen from of Eq. (21) for a given value of α. The calculated ρ Fig. 1, the right branch of isocharge for Schw-type values of the critical quantities, such as qc, Tc, þc, black hole is locally stable, i.e., the specific heat at and Sc,forvariousα are illustrated in Figs. 2 and 3.

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(a) (b)

α FIG. 2. The behaviors of the critical electric charge (qc) and criticalpffiffiffi temperature (Tcp)ffiffiffi versus . The no BH region corresponds to no BH solution. The vertical dashed lines mark the values of α ¼ 1= 3, α ¼ 1, and α ¼ 3. We use the logarithmic scales on the vertical axis and set l ¼ 1.

(a) (b)

ρ α FIG. 3. The behaviorspffiffiffi of the criticalp eventffiffiffi horizon radius ( þc) and critical entropy (Sc) versus . The vertical dashed lines mark the values of α ¼ 1= 3, α ¼ 1 and α ¼ 3. We use the logarithmic scales on the vertical axis and set l ¼ 1.

We observe that there is no critical point for charged In order to fully obtain phase transition and dilaton AdS black hole in cases where α 1 and examine phase structure of charged dilaton AdS pffiffiffi ¼ α ≥ 3 ≈ 1.73.AsonecanseefromFigs.2 and 3, black holes, we shall study the behavior of Helmholtz for 0 < α < 1, the dilaton coupling parameter (α) free energy in the next section. does not significantly affect the critical quantities, except entropy which abruptly decreases close to 1. IV. HELMHOLTZ FREE ENERGY Figure 2(b) shows that the critical point occurs in the pffiffiffi The general thermodynamic description of charged RN-type of black hole when 0 ≤ α < 1= 3,whereas pffiffiffi dilaton AdS black hole is provided by studying the for 1= 3 < α < 1 it occurs in Schw-type where Helmholtz free energy which exhibits the global stable there is a lower bound on temperature of the black state. Indeed, the Helmholtz free energy is an appro- hole. As expected from Figs. 2 and 3,intheabsence priate thermodynamic potential to characterize equilib- α 0 of dilaton field ( ¼ ), the critical quantities reduce rium processes at constant temperature and charge in to those of charged AdS black hole [4]. In case of pffiffiffi the canonical ensemble [28]. The Helmholtz free 1 < α < 3, with increasing α, the values of critical pffiffiffi energy for a fixed AdS radius regime can be obtained quantities increase and diverge for α → 3. It should pffiffiffi through the Legendre transformation of the mass M. also be pointed out that in the range 1 < α < 3,the Thus, the Helmholtz free energy per unit volume ω is critical behavior happens in Schw-type region. given

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0.014 0.013 0.012 0.012

0.011 0.010

0.010 0.008 0.009 0.006 0.008 0.24 0.25 0.26 0.27 0.28 0.05 0.10 0.15 0.20 0.25 0.30

(a) (b)

FIG. 4. Helmholtz free energy as a function of temperature for l ¼ 1 and various values of q.Forq

HðT;qÞ¼M −TS figures, it can be seen that the charge dependence of Helmholtz free energy is strongly affected by the dilaton ϒ 3−4Γ−α2 2 α2 −1 α2 5 pffiffiffi lð ffiffiffi½ þ ð Þþ þ Þ ¼ p 2 ; ð22Þ coupling parameter α. In case of α ¼ 0.5 ∈ ½0; 1= 3, 32 2π α2 1 Γ3=2−2=ðα þ1Þ ϒ−1 −1=2 ð þ Þ ð Þ where charged dilaton AdS black hole is RN-type, the Helmholtz free energy is single value in temperature for where q>qc [see Fig. 4(a)]. In this case, back hole is locally sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi stable (Cq > 0) which is indicated by the solid blue line in 2 2 3−4= α2 1 4q ðα þ 1ÞΓ ð þ Þ Fig. 4(a). On the other hand, when qb) dashed-red line in Fig. 4(a). A close up of such a curve is leads to the constraint 0 < Γ < 1. illustrated in the inset of Fig. 4(a), where the arrows The behavior of Helmholtz free energy in terms of the indicate the direction of the decreasing black hole event horizon (ρ ). Decreasing the temperature of the system temperature T for α ¼ 0.5, 0.7, and 1.3 are depicted in þ Figs. 4 and 5, for different values of charge q. From these follows the lower solid blue curve, which corresponds to the lowest Helmholtz free energy, until it crosses the upper solid blue curve. In this position, system enters to the 2 left solid blue curve with a first order large black hole 2.0 (LBH)/small black hole (SBH) phase transition which is 1.5 accompanied by a discontinuity in the slop of Helmholtz free energy.1 The small (large) is referred to the size of 1.5 1 pffiffiffi ρ α 0 7 ∈ 1= 3; 1 H 0.5 1 1.5 2 horizon radius þ.For ¼ . ð Þ case, as illustrated in Fig. 4(b), charged dilaton AdS black hole q qc 1.0 is Schw-type where the lower (upper) branch of the q q Helmholtz free energy is thermodynamically stable (unsta- c 0 0 ble) with Cq > (Cq < ) for q>qc.Atq ¼ qc, there is a q qc 0.5 critical point in the lower branch of H.Forqqc, the system undergoes a first order phase transition between SBH and 1 LBH. The positive (negative) sign of Cq is identified by the blue A first order phase transition occurs when there exist a solid (dashed red) line. Inset: Black arrows show the direction of discontinuity in the first derivative of Helmholtz free energy ρ ∂ ∂ the decreasing þ. The curves are shifted for clarity. with respect to temperature which is entropy, i.e., ð H= TÞ¼S.

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FIG. 6. SBH/LBH phase diagram for l ¼ 1 and various values of α. The critical points and first order phase transition curves are highlighted by the black solid circle and solid orange line, respectively. At low temperature, no BH regions correspond to no black hole solution. We use the logarithmic scale on q axis in Fig. (b). decreases along the lowest stable (solid blue) branch of the V. SUMMARY curve until two stable branches cross each other. At this To sum up, we have revisited critical behavior and phase point, black hole enters to the solid blue branch with the structure of charged dilaton black holes in the background discontinuous change in the slope of Helmholtz free of AdS spaces. The motivation for study phase behavior of energy, indicating a first order phase transition which dilaton black holes in AdS spacetime is mainly inspired by occurs between the LBH and SBH. Further decreasing AdS=CFT correspondence and is expected to shed light on T, black hole follows the left solid blue branch until the pffiffiffi the microscopic structure of black holes. We adopted the end. For α ¼ 1.3 ∈ ð1; 3Þ, a novel behavior happens in view point that cosmological constant can be regarded as a for Schw-type black hole (see Fig. 5). Indeed, in contrast to fixed parameter, while the charge of the black hole varies. what occurs in Fig. 4(b), in this case a first order phase To understand the impact of the dilaton field on the heat transition takes place between SBH and LBH for q>qc.In capacity, Cq, which determines the local thermodynamic ρ the inset of Fig. 5, the decrease in þ is denoted by the stability of the system, we have studied the behavior of the arrows, which is opposite to the previous case, i.e., the inset temperature T as a function of entropy S for different values in Fig. 4(b). This behavior has not been observed in of α. By expanding T for small values of S,wehave previous studies on phase transition of charged AdS black distinguished several black hole systems, with thermal holes [26,33]. It is notable to mention that we do not find stability/instability, depending on the values of α.Inorder any other phase transition for charged dilaton AdS to obtain the coordinates of the critical point, we numerically black hole. solved the system of equations and plotted the quantities at The corresponding SBH/LBH phase diagram of dilaton the critical point in terms of α. We observed that there is no pffiffiffi AdS black hole for different values of the dilaton critical point in cases with α ¼ 1 and α ≥ 3, while for parameter α is sketched in Fig. 6.Itisclearfrom 0 < α < 1, the critical quantities are not significantly Fig. 6 that the critical points are denoted by black spots affected by α, except entropy which abruptly decreases at the end of the first order phase transition curves close to 1. In the absence of dilaton field (α ¼ 0), the critical (orange). In Fig. 6(a), the first order phase transition quantities reduce to those of charged AdS black hole. curves separate the SBH from LBH for qqc. Also, no BH regions implies that no BH values of α and q. We have realized several cases, depend- solutions exist at the low temperature. ing on α, including whether or not the Helmholtz free

064021-7 AMIN DEHYADEGARI and AHMAD SHEYKHI PHYS. REV. D 102, 064021 (2020) energy is single/multi-valued and whether or not the system WðRÞ¼fðρðRÞÞ; is thermally stable/unstable. Interestingly enough, we   observed that unstable behavior of the Helmholtz free dR 2 VðRÞ¼fðρðRÞÞ energy for qqc. The later has not been observed in the previous studies on phase where f is given by Eq. (4). According to the metric transition of charged AdS black holes and is one of the new Eq. (4), the background metric is chosen to be of the form result of the present work. W0ðRÞ¼V0ðRÞ¼f0ðρðRÞÞ ACKNOWLEDGMENTS ρ2 2α2bρ α4b2 ¼ 1 þ − þ : We are grateful to the Research Council of Shiraz l2 l2ðα2 þ 1Þ l2ðα2 þ 1Þ2 University. This work has been supported financially by Research Institute for Astronomy and Astrophysics of To compute the mass of the space time, one should choose a Maragha, Iran. A. S. thanks Hermann Nicolai and Max- timelike Killing vector ξ on the boundary surface B of the Planck Institute for Gravitational Physics (AEI), for space time Eq. (A1). Then, the quasilocal mass can be hospitality. calculated via Z 1 pffiffiffi APPENDIX: MASS OF DILATON BLACK HOLE 2φ σ − − 0 − 0 0 aξb M ¼ d fðKab KhabÞ ðKab K habÞgn ; IN AdS SPACE 8π B To obtain the mass of a charged dilaton AdS black hole ðA2Þ in four dimensions, we use the background subtraction σ σ method of the Brown-York which is based on quasilocal where is the determinant of the metric ab of the boundary B, na is the timelike unit normal vector to the concept [23,38]. Within this formalism, the line element 0 should be written in the form boundary and Kab is the extrinsic curvature of the back- ground metric. In the context of the counterterm method, 2 dR the limit in which the boundary becomes infinity, B∞,is ds2 ¼ −WðRÞdt2 þ þ R2dΩ2: ðA1Þ VðRÞ taken, and the counterterm prescription ensures that the action and mass are finite. By using the above modified In order to obtain the metric Eq. (A1) from Eq. (3),we Brown-York formalism, one can obtain the mass of the AdS perform the following transformation dilaton black hole per unit volume as     γ 2 1 α2 − 1 b = − R ¼ ρ 1 − ; M ¼ c b 2 : ðA3Þ ρ 8π α þ 1 such that, the metric functions W and V are written as In the absence of the dilaton (α ¼ 0), this expression reduces to the mass of the AdS black hole in four dimensions.

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